The first type of motion we have learned in linear kinematics wasunder a constant acceleration. We will learn about the rotationalmotion under constant angularacceleration (α),because these arethe simplest motions in both cases.

Just like the case in linear motion, one can obtain

Angularvelocity

under constantangular acceleration:

Angular displacement underconstant angular acceleration:

fOne can also obtain

f200022f0t0t12t2vLinear kinematics

Linear kinematics

fx2102ox v t at Linear kinematics

2fv22o f iv a x x ov atWednesday, April 10,2013

PHYS 1441-002, Spring 2013Dr. Jaehoon Yu

4

Rotational Kinematics ProblemSolving Strategy

•Visualize the problem by drawing a picture.

•Write down the values that are given for any of thefive

kinematic variables

and convert them to SI units.

–Remember that the unit of the angle must bein radians!!

•Verify that the information contains values for at leastthree

of the five kinematic variables. Select theappropriate equation.

•When the motion is divided into segments, rememberthat

the final angular

velocity of one segment is theinitialvelocity

for the next.

•Keep in mind that there may be two possible answersto a kinematics problem.

Wednesday, April 10,2013

PHYS 1441-002, Spring 2013Dr. Jaehoon Yu

5

Example for Rotational Kinematics

A wheel rotates with a constant angular acceleration of 3.50 rad/s2. Ifthe angular speed of the wheel is 2.00 rad/s atti=0, a) through whatangle does the wheel rotate in 2.00s?

fiUsing the angular displacement formula in the previous slide, one gets

t2.002.00123.502.00211.0rad11.02rev.1.75rev.12t2Wednesday, April 10,2013

PHYS 1441-002, Spring 2013Dr. Jaehoon Yu

6

Example for Rotational Kinematicscnt’d

What is the angular speed at t=2.00s?

fitUsing the angular speed and acceleration relationship

Find the angle through which the wheel rotates between t=2.00 sand t=3.00 s.

2.002.00123.502.0011.0rad2.003.00123.503.00221.8radWednesday, April 10,2013

PHYS 1441-002, Spring 2013Dr. Jaehoon Yu

7

The blade is whirling with an angular velocity of +375rad/s when

the“puree”

button is pushed in.

When the“blend”

button is pushed,

the blade accelerates andreaches a

greater angular velocity after the blade hasrotated through an

angular displacement of +44.0 rad.

The angular acceleration has a

constant value of +1740rad/s2.

Find the final angular velocity of the blade.

Ex. Blending with a Blender

θ

α

ω

ωo

t

2 22o   Which kinematic eq?

o22375rads221740rads244.0rad542radsWhich sign?

542rad sWhy?

Because the blade is accelerating in counter-clockwise!

+44.0rad

+375rad/s

?

+1740rad/s2

Wednesday, April 10,2013

PHYS 1441-002, Spring 2013Dr. Jaehoon Yu

8

Relationship Between Angular and Linear Quantities

What do we know about a rigid object that rotatesabout a fixed axis of rotation?

When a point rotates, it has both the linear and angularcomponents in its motion.

What is the linear component of the motion you see?

vEvery particle (ormasslet) inan objectmoves in a circle centeredat the same axis of rotation with the same angular velocity.

Linear

velocity along the tangential direction.

How do we related this linear component of the motionwith angular component?

lrThe arc-length is

So the tangential speedv

is

What does this relationship tell you aboutthe tangential speed of the points in theobject and their angular speed?

Although every particle in the object has the sameangular speed, its tangential speed differs and isproportional to its distance from the axis of rotation.

The farther away the particle is from the center ofrotation, the higher the tangential speed.

Thedirectionof

follows theright-handrule.

lt

rt

rtrWednesday, April 10,2013

PHYS 1441-002, Spring 2013Dr. Jaehoon Yu

9

Isthelion faster thanthehorse?

A rotating carousel has one child sitting onthehorse near the outer edgeand another child onthelion halfway out from the center. (a) Which childhas the greater linear speed? (b) Which child has the greater angularspeed?

(a)Linear speed is the distance traveleddivided by the time interval. So the childsitting at the outer edge travels moredistance within the given time than the childsitting closer to the center. Thus, the horseis faster than the lion.

(b) Angular speed is the angle traveled divided by the time interval. Theangle both the children travel in the given time interval is the same.Thus, both the horse and the lion have the same angular speed.

Wednesday, April 10,2013

PHYS 1441-002, Spring 2013Dr. Jaehoon Yu

10

How about the acceleration?

vtTwo

How many different linear acceleration components doyou see in a circular motion and what are they?

Total linear acceleration is

Since the tangential speedv

is

What does thisrelationship tell you?

Although every particle in the object has the same angularacceleration, its tangential acceleration differs proportional to itsdistance from the axis of rotation.

Tangential,at, and the radial acceleration,ar.

atThe magnitude of tangentialaccelerationat

is

The radial or centripetal accelerationar

is

raWhat doesthis tell you?

The father away the particle is from the rotation axis, the more radialacceleration it receives. In other words, it receives more centripetal force.

Torque is the tendency of a force to rotate an object about an axis.Torque,, is a vector quantity.

Magnitude of torque is defined as the product of the forceexerted on the object to rotate it and the moment arm.

F



l1

The lineof Action

Consider an object pivoting about the pointP

bythe forceFbeing exerted at a distancerfrom

P.

P

r

Moment arm

The line that extends out of the tail of the forcevector is called theline of action.

The perpendicular distance from the pivoting pointP

to theline of action

is calledthe moment arm.

When there are more than one force being exerted on certainpoints of the object, one can sum up the torque generated by eachforcevectorially. The convention for sign of the torque ispositiveif rotation is in counter-clockwise